Whisk together egg, salt, and pepper. Using a grater, shred zucchini, carrot, and potato. Combine vegetables with egg mixture, 1/2 cup basil and flour.
Heat oil in a skillet over medium-high heat. Spoon batter into pan in 1/4 cup portions, 3 or 4 at a time; level out with a spatula or back of a measuring cup, pancakes should be about 3 1/2 inches round. Cook pancakes 6 to 8 minutes, flipping a few times so both sides are evenly golden. Move pancakes to a platter and place in a 200 degree F oven. Continue cooking until all pancakes are made.
Combine sour cream with 2 tablespoons basil. Serve pancakes piping hot with sour cream.